WASHINGTON — Democrats will take control of both the White House and Congress on Wednesday for the first time in a decade, but they have already discovered the difficulties of governing with so little room to maneuver around Republicans who appear in no mood to cooperate.

Even before President-elect Joseph R. Biden Jr. was sworn in as the new president, a Republican senator put up a roadblock on Tuesday to one of his cabinet nominees, helping deny Mr. Biden the Day 1 confirmation of national security officials that other incoming presidents have taken for granted for decades. And Senator Chuck Schumer, the New York Democrat who is set to become majority leader, found himself in difficult negotiations with Senator Mitch McConnell, the Kentucky Republican and incoming minority leader, over plans for running an evenly split Senate.

On top of those hardly trifling issues, the looming impeachment trial of President Trump presents an extraordinary complication for a party that would no doubt prefer to focus on a bold Democratic agenda out of the gate. However it is eventually structured, the trial is certain to sap time, energy and momentum from the opening days of the new Democratic era in Washington.

“It does put a wrinkle in the road,” conceded Christopher J. Dodd, a former Democratic senator from Connecticut and a close adviser to Mr. Biden. “I know Joe is disappointed that it will take some time. But the fact of the matter is, what occurred that day was of such significance that to ignore it or treat it in a casual way would only be an invitation for it to happen again.”

The Republican resistance comes as Mr. Biden has made clear that he wants to work with both parties to achieve his legislative ambitions and still sees an opportunity for bipartisan consensus in a Senate that has become much more polarized since he left in 2009 after 36 years.

Mr. Schumer is set to become majority leader by the narrowest possible margin on Wednesday afternoon after the inauguration of Mr. Biden and Vice President-elect Kamala Harris and the seating of three new Democratic senators, setting the margin at 50 to 50 with Ms. Harris empowered to cast tiebreaking votes.

Dismissing concerns about the implications of the Senate trial for the Biden agenda, Mr. Schumer and his colleagues have said they can manage to do two things at once, conducting the proceeding while advancing Mr. Biden’s nominees and his other priorities.

But the Senate has struggled in recent years to do even one thing at once, and experience has shown that presidential impeachments consume a lot of oxygen.

Still, Democrats say they have no choice and few reservations about plunging ahead with a trial they see as a crucial element of a national reckoning after the violent Jan. 6 assault on the Capitol and attempt to disrupt the counting of the presidential Electoral College ballots.

“We have to send a very strong message that it is unacceptable for a president of the United States to incite a violent mob for the purposes of overturning a democratic election,” said Senator Chris Van Hollen, Democrat of Maryland. “I think we can make clear that this conduct has to be held accountable while we can work to open a new chapter.”

Mr. Schumer, Mr. Van Hollen and other Democrats insist that the trial of Mr. Trump for inciting the attack can be conducted quickly if Republicans cooperate. They were buoyed on Tuesday by a strong statement from Mr. McConnell, who made clear that he held the president responsible for the violence and said Mr. Trump had “provoked” the mob.

But even as Mr. McConnell has indicated an openness to allowing the trial to move forward, he was playing hardball in talks with Mr. Schumer on a so-called power sharing arrangement for managing the Senate in the longer term. In a memo to Republican colleagues and in talks with Mr. Schumer, Mr. McConnell said he wanted Democrats to agree not to weaken the filibuster in exchange for his cooperation in cutting a deal that would let Senate committees get down to business.

That created a dilemma for Mr. Schumer, who, along with Democratic activists, wants to hold out the idea of eliminating the 60-vote threshold for advancing legislation — which in recent years has allowed the minority party to block most major initiatives — if Republicans uniformly line up against Mr. Biden’s agenda. The Democratic leader’s office urged Mr. McConnell to drop his demand and instead implement a plan developed 20 years ago, when the Senate was evenly divided and the two parties split committees down the middle.

“Leader Schumer expressed that the fairest, most reasonable and easiest path forward is to adopt the 2001 bipartisan agreement without extraneous changes from either side,” said Justin Goodman, a spokesman for Mr. Schumer.

Regardless of how the negotiations go, Mr. McConnell made it clear Tuesday that Republicans had no intention of clearing the way for Democrats’ most cherished progressive priorities.

“Certainly November’s elections did not hand any side a mandate for sweeping ideological change,” he said. “Americans elected a closely divided Senate, a closely divided House and a presidential candidate who said he’d represent everyone.”

Mr. Biden’s push to have the Senate confirm some of his key nominees as soon as he took the oath Wednesday encountered an obstacle when Senator Josh Hawley, the Missouri Republican who led the challenge to the Electoral College vote, said he would block quick confirmation of Mr. Biden’s secretary of homeland security nominee, Alejandro N. Mayorkas, over immigration concerns.

“Mr. Mayorkas has not adequately explained how he will enforce federal law and secure the southern border given President-elect Biden’s promise to roll back major enforcement and security measures,” Mr. Hawley said.

His move infuriated Democrats, many of whom blame Mr. Hawley for the Jan. 6 riot that prompted the heightened security concerns that they say require the position to be filled immediately.

“I think it really would be disgraceful, and shameful, but unfortunately, Senator Hawley is marching to his own drummer,” said Senator Richard Blumenthal, Democrat of Connecticut, about the delay.

The contours of the impeachment trial remain unclear, and even the start date is unknown since Speaker Nancy Pelosi has not yet presented the Senate with the sole charge. Democrats would like to operate on two tracks and conduct Senate business for part of the day while holding the trial during the remainder — an approach that would require discipline and long hours.

But Republicans, some of whom say Democrats have boxed themselves in by insisting on the trial during the crucial opening days of Mr. Biden’s tenure, offered a reminder and a warning Tuesday that impeachment takes center stage in the Senate once a trial begins.

“Once she sends the articles of impeachment over, it displaces all other business,” Senator John Cornyn, Republican of Texas, told reporters on Capitol Hill.

While the Senate obviously has its hands full and the new era is not off to the smoothest start, Mr. Schumer, rising to a position he has coveted for years, said Democrats had big plans for their majority.

“The next several months will be very busy,” he said, “and a very consequential period for the United States Senate.”